This article introduces Gamma-triangles, which are closely related to and more fundamental than F-triangles and H-triangles that have been used in the combinatorics of cluster complexes. It is proved that Gamma-triangles can be expressed as sums of the local gamma-vectors introduced by Athanasiadis, which themselves refine the local h-vectors attached by Stanley to simplicial subdivisions. Gamma triangles of all cluster complexes are then explicitly computed.
We characterize a family of number triangles whose production matrices are closely related to the original number triangle. We study a number of such triangles that are of combinatorial significance. For a specific subfamily, these triangles relate to sequences that have interesting convolution recurrences and continued fraction generating functions.
We prove that the maximum number of triangles in a $C_5$-free graph on $n$ vertices is at most $\frac{1}{2 \sqrt 2} (1 + o(1)) n^{3/2}$, improving an estimate of Alon and Shikhelman.
In their unpublished work, Jockusch and Propp showed that a 2-enumeration of antisymmetric monotone triangles is given by a simple product formula. On the other hand, the author proved that the same formula counts the domino tilings of the quartered Aztec rectangle. In this paper, we explain this phenomenon directly by building a correspondence between the antisymmetric monotone triangles and domino tilings of the quartered Aztec rectangle.
This is a companion note to the paper "Almost all Steiner triple systems have perfect matchings (arXiv:1611.02246). That paper contains several general lemmas about random Steiner triple systems; in this note we record analogues of these lemmas for random Latin squares, which in particular are necessary ingredients for our recent paper "Large deviations in random Latin squares" (arXiv:2106.11932). Most important is a relationship between uniformly random order-$n$ Latin squares and the triangle removal process on the complete tripartite graph $K_{n,n,n}$.
We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes $\infty,0,1$. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained.
Let $G$ be a triangle-free graph on $n$ vertices with adjacency matrix eigenvalues $μ_1(G)\geq μ_2(G)\geq \dots \geq μ_n(G)$. In this paper we study the quantity $$μ_1(G)+μ_n(G).$$ We prove that for any triangle-free graph $G$ we have $$μ_1(G)+μ_n(G)\leq (3-2\sqrt{2})n.$$ This was proved for regular graphs by Brandt, we show that the condition on regularity is not necessary. We also prove that among triangle-free strongly regular graphs the Higman-Sims graph achieves the maximum of $$\frac{μ_1(G)+μ_n(G)}{n}.$$
The suspension of the path $P_4$ consists of a $P_4$ and an additional vertex connected to each of the four vertices, and is denoted by $\hat{P_4}$. The largest number of triangles in a $\hat{P_4}$-free $n$-vertex graph is denoted by $ex(n,K_3,\hat{P_4})$. Mubayi and Mukherjee in 2020 showed that $ ex(n,K_3,\hat{P_4})= n^2/8+O(n)$. We show that for sufficiently large $n$, $ex(n,K_3,\hat{P_4})=\lfloor n^2/8\rfloor$.
Let $α_{1} (G)$ denote the maximum size of an edge set that contains at most one edge from each triangle of $G$. Let $τ_{B} (G)$ denote the minimum size of an edge set whose deletion makes $G$ bipartite. It was conjectured by Lehel and independently by Puleo that $α_{1} (G) + τ_{B} (G) \le n^2/4$ for every $n$-vertex graph $G$. Puleo showed that $α_{1} (G) + τ_{B} (G) \le 5n^2/16$ for every $n$-vertex graph $G$. In this note, we improve the bound by showing that $α_{1} (G) + τ_{B} (G) \le 4403n^2/15000$ for every $n$-vertex graph $G$.
In this short note, we give a lower bound on the number of congruence classes of triangles in a small set of points in $\mathbb{F}_p^2$. More precisely, for $\mathcal{A}\subset \mathbb{F}_p^2$ with $|\mathcal{A}|\le p^{2/3}$, we prove that the number of congruence classes of triangles determined by points in $\mathcal{A}\times \mathcal{A}$ is at least $|\mathcal{A}|^{7/2}$. This note is not intended for journal publication.
The random greedy algorithm for constructing a large partial Steiner-Triple-System is defined as follows. We begin with a complete graph on $n$ vertices and proceed to remove the edges of triangles one at a time, where each triangle removed is chosen uniformly at random from the collection of all remaining triangles. This stochastic process terminates once it arrives at a triangle-free graph. In this note we show that with high probability the number of edges in the final graph is at most $ O\big( n^{7/4}\log^{5/4}n \big) $.
In this note we investigate some matroid minor structure results. In particular, we present sufficient conditions, in terms of {\em triangles}, for a matroid to have either $U_{2,4}$ or $F_7$ or $M(K_5)$ as a minor.
A set of $q$ triangles sharing a common edge is a called a book of size $q$. Letting $bk(G)$ denote the size of the largest book in a graph $G$, Erdős and Rothschild \cite{erdostwo} asked what the minimal value of $bk(G)$ is for graphs $G$ with $n$ vertices and a set number of edges where every edge is contained in at least one triangle. In this paper, we show that for any graph $G$ with $n$ vertices and $\frac{n^2}{4} - nf(n)$ edges where every edge is contained in at least one triangle, $bk(G) \geq Ω\left(\min{\{\frac{n}{\sqrt{f(n)}}, \frac{n^2}{f(n)^2}\}}\right)$.
This note collects some facts and conjectures about the Hankel determinants and their generating functions of the columns of Hoggatt triangles which apparently are related to combinatorial objects such as Young tableaux and Narayana numbers.
In this short note we present a set of interesting and useful properties of a one-parameter family of sequences including factorial and subfactorial, and their relations to the Gamma function and the incomplete Gamma function.
B. D. Acharya has conjectured that if $\bigl(A_i: i=1, 2, ..., 2^{|X|}-1\bigr)$ is a permutation of all nonempty subsets of a set $X$ with at least two elements such that for each even positive integer $j<2^{|X|}-1$, $A_{j-1}\triangle A_j\triangle A_{j+1}=\emptyset$, then $|X|=2$. In this article, we show that if the cardinality of a set $X$ is more than four, then a permutation as described above indeed exists.
In this paper, we deal with the signed bad number and the negative decision number of graphs. We show that two upper bounds concerning these two parameters for bipartite graphs in papers [Discrete Math. Algorithms Appl. 1 (2011), 33--41] and [Australas. J. Combin. 41 (2008), 263--272] are not true as they stand. We correct them by presenting more general bounds for triangle-free graphs by using the classic theorem of Mantel from the extremal graph theory and characterize all triangle-free graphs attaining these bounds.
Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Grötzsch's result that triangle-free planar graphs are such, most of the effort was focused to solving Havel's and Steinberg's conjectures. In this paper, we prove that every planar graph of maximum degree $4$ obtained as a subgraph of the medial graph of any bipartite plane graph is $3$-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence represen…
A note on vertex-critical induced subgraphs of shift graphs
University of West Bohemia（西波希米亚大学）；CNRS（法国国家科学研究中心）；IRIF（IRIF实验室）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Faculty of Information Studies in Novo Mesto（诺沃梅斯托信息研究学院）；Rudolfovo – Science and Technology Centre（鲁多维福科学与技术中心）；University of Southern Denmark（南丹麦大学）
A book of size N is the union of N triangles sharing a common edge. We show that the Ramsey number of a book of size N vs. a book of size M equals 2N+3 for all N>(10^6)M. Our proof is based on counting.
In this note we prove an assertion made by M. Levin in 1999: the Pascal matrix modulo 2 has the property that each of the square sub-matrices laying on the upper border or on the left border has determinants, computed in $\mathbb{Z}$, equal to 1 or -1.
We show the existence of a series of transforms that capture several structures that underlie higher-dimensional partitions. These transforms lead to a sequence of triangles whose entries are given combinatorial interpretations as the number of particular types of skew Ferrers diagrams. The end result of our analysis is the existence of a triangle, that we denote by F, which implies that the data needed to compute the number of partitions of a given positive integer is reduced by a factor of half. The number of spanning rooted forests appears intriguingly in a family of entries in the triangle…
The Appendix of the article arXiv:1212.5056 [math.CO] "On growth in an abstract plane" by Nick Gill, H. A. Helfgott, Misha Rudnev, contains a general "geometric Ruzsa triangle inequality" in a Desarguesian projective plane. The purpose of this note is to give a similar inequality for metric spaces with dilations, that is in the absence of an algebraic or incidence structure.
In this short note, we give a localized version of the basic triangle theorem, first published in 2011 (see [4]) in order to prove the independence of hyperlogarithms over various function fields. This version provides direct access to rings of scalars and avoids the recourse to fraction fields as that of meromorphic functions for instance.
We give the connection between three polynomials that generate triangles in The On-Line Encyclopedia of Integer Sequences (A123192, A137396 and A300453). We show that they are related with the bracket polynomial for the (2,n)-torus knot
A family $F$ of graphs on a fixed set of $n$ vertices is called triangle-intersecting if for any $G_1,G_2 \in F$, the intersection $G_1 \cap G_2$ contains a triangle. More generally, for a fixed graph $H$, a family $F$ is $H$-intersecting if the intersection of any two graphs in $F$ contains a sub-graph isomorphic to $H$. In [D. Ellis, Y. Filmus, and E. Friedgut, Triangle-intersecting families of graphs, J. Eur. Math. Soc. 14 (2012), pp. 841--885], Ellis, Filmus and Friedgut proved a 36-year old conjecture of Simonovits and Sós stating that the maximal size of a triangle-intersecting family is…
We prove that there is an absolute constant $C>0$ so that for every natural $n$ there exists a triangle-free \emph{regular} graph with no independent set of size at least $C\sqrt{n\log n}$.
Algoresearch L.L.C.（Algoreresearch有限责任公司）；Research Institute of the University of Bucharest（布加勒斯特大学研究所）；Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）
